3(n^2+8n+8)=0

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Solution for 3(n^2+8n+8)=0 equation: 


Simplifying
3(n2 + 8n + 8) = 0

Reorder the terms:
3(8 + 8n + n2) = 0
(8 * 3 + 8n * 3 + n2 * 3) = 0
(24 + 24n + 3n2) = 0

Solving
24 + 24n + 3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(8 + 8n + n2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(8 + 8n + n2)' equal to zero and attempt to solve:

Simplifying
8 + 8n + n2 = 0

Solving
8 + 8n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 8n + -8 + n2 = 0 + -8

Reorder the terms:
8 + -8 + 8n + n2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 8n + n2 = 0 + -8
8n + n2 = 0 + -8

Combine like terms: 0 + -8 = -8
8n + n2 = -8

The n term is 8n.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8n + 16 + n2 = -8 + 16

Reorder the terms:
16 + 8n + n2 = -8 + 16

Combine like terms: -8 + 16 = 8
16 + 8n + n2 = 8

Factor a perfect square on the left side:
(n + 4)(n + 4) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(n + 4) equal to 2.828427125 and -2.828427125.


Subproblem 1
n + 4 = 2.828427125

Simplifying
n + 4 = 2.828427125

Reorder the terms:
4 + n = 2.828427125

Solving
4 + n = 2.828427125

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-4' to each side of the equation.
4 + -4 + n = 2.828427125 + -4

Combine like terms: 4 + -4 = 0
0 + n = 2.828427125 + -4
n = 2.828427125 + -4

Combine like terms: 2.828427125 + -4 = -1.171572875
n = -1.171572875

Simplifying
n = -1.171572875


Subproblem 2
n + 4 = -2.828427125

Simplifying
n + 4 = -2.828427125

Reorder the terms:
4 + n = -2.828427125

Solving
4 + n = -2.828427125

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-4' to each side of the equation.
4 + -4 + n = -2.828427125 + -4

Combine like terms: 4 + -4 = 0
0 + n = -2.828427125 + -4
n = -2.828427125 + -4

Combine like terms: -2.828427125 + -4 = -6.828427125
n = -6.828427125

Simplifying
n = -6.828427125


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-1.171572875, -6.828427125}
Solution
n = {-1.171572875, -6.828427125}

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